I have the following utility maximization problem: $$\max (xy)$$ $$(x+y-2)^2 \leq 0$$ Conditions: $$y-2\lambda (x+y-2) =0$$ $$x-2\lambda (x+y-2) =0$$ $$\lambda(x+y-2)^2=0$$
When I set $\lambda>0$, I get: $$(x+y-2)^2=0 \Rightarrow (x+y-2) = 0$$ $$y-2\lambda (x+y-2) = y = 0$$ $$x-2\lambda (x+y-2) = x = 0$$
But the obvious solution is $x=y=1$.
When I set $$\lambda=0$$ that's not a case with a valid solution either. Is the constraint too low? What is the explanation to this?
I would greatly appreciate your help.